---
tag: JTVWRXXAKM
kind: concept
name: Normal family
msc: [30D45]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_family
same_as:
  wikidata: Q1049280
  wikipedia: Normal family
---

## Summary

A normal family is a family of holomorphic functions in which every sequence has a subsequence that converges uniformly on compact sets, see the [[definition|QWSD2JDRP0]].
It is the complex-analytic version of a relatively compact set of functions.
[[Montel's theorem|ZVFVBVJMCB]] gives simple tests for normality.
Normal families are a key tool in the proof of the [[Riemann mapping theorem|02WHMEG5ZR]] and in complex dynamics.

## Explanations

### L1 human

In a finite set, every sequence repeats some element infinitely often.
A normal family is a family of functions with a similar property: from any sequence one can pick a subsequence that settles down to a limit.
The limit is reached uniformly on each closed and bounded piece of the domain, though maybe not on the whole domain.
For example, the powers $z, z^2, z^3, \dots$ on the unit disc form a normal family, since they tend to $0$ on each smaller disc.